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How do I solve the following question.

The repeated decimal expansion $1.23\overline 6$ corresponds to which fraction?

a. $\frac{370}{300}$ b. $\frac{3710}{3001}$ c. $\frac{371}{301}$ d. $\frac{37100}{30001}$ e. $\frac{371}{300}$

Here is how I am trying to do but I get the wrong answer.

$100α − α = 123.6 − 1.236 = 122.364$

Hence $99α = 122.364 \Rightarrow α = \frac{122.364}{99}$

Thank you.

callculus42
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jesse
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    Do you mean $S=1.\overline {236}$? If so, consider $1000S-S$. – lulu Aug 04 '19 at 15:06
  • Or you probably mean $1.23\overline 6$ – callculus42 Aug 04 '19 at 15:10
  • Hint: $ 1.23\bar6\times100=123\dfrac23$; or what is $1000a-100a?$ – J. W. Tanner Aug 04 '19 at 15:11
  • You must tell us what is being repeated. I had to assume it was $236$ but that leads to none of the solutions. – fleablood Aug 04 '19 at 15:13
  • $3x = 3.71,$ so $,300x = \ldots \ \ $ – Bill Dubuque Aug 04 '19 at 15:17
  • Jesse, I´ve made an edit. Please check if my modifications are right. And do the hints help? – callculus42 Aug 04 '19 at 15:23
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    Your mistake was that you didn't actually carry any of the repeating $6$s. You just did $1.236$ and not $1.2366666666666.....$. When you multiplied by $100$ you got $123.6 = 123.600$ and not $123.66666666....$ and you subtracted $1.236$ and not $1.236666666.....$. So $100a - a = 123.600 - 1.236$ will tell you what $1.236$ is as a fraction. but $123.666- 1.236$ will tell you what $1.2366666...$ is as a fraction. (Also you don't need to multiply by 100. You can multiply by just $10$). – fleablood Aug 04 '19 at 15:47

6 Answers6

5

Working backwards, try some of those divisions.

$371 / 300 = 1.23 \overline{6}$

badjohn
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5

It's multiple choice. In a test situation with a clock ticking down, sometimes you just have to pick the answer that looks the most sensible to you, hope for the best and move on.

But if this is homework, you can just put the five choices through a calculator at your leisure, plus other similar cases to help solidify your understanding.

Thus you would see that $$\frac{370}{300} \approx 1.23333333$$ The wrong digit repeats. Can the numerator be even? Maybe, but then the fraction wouldn't be in lowest terms... unless the denominator is odd, like in the next choice: $$\frac{3710}{3001} \approx 1.23625458$$ That doesn't seem to have any consecutive repeated digits.

Something tells me that in base 10, the only way to get an infinitely repeating 3 or 6 is for the denominator to be a nonzero multiple of 3, and for the numerator to not be a multiple of 3.

Clearly 3001 is not a multiple of 3, and in fact it's prime. 301 is not prime, but it's not a multiple of 3 either. 30001 is not prime either, but it is also quite obviously not a multiple of 3.

This leaves $$\frac{371}{300} \approx 1.23666667$$ Clearly we had a loss of machine precision there, but this has got to be the right answer.

J. W. Tanner
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Robert Soupe
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4

Since you're interested in making this more complicated than it needs to be, did you try multiplying $1.23666$ by each of the denominators? An approximation is good enough for the purpose here, no need to test how many trailing digits you can punch up on your calculator. I like six $6$s. Mwahahahaha!

Might as well start with a. Verify that $1.23666 \times 300 = 370.998$. This overshoots $370$ by almost $1$, suggesting that the answer is $371$ divided by $300$, but then that would make for a short episode.

So instead we try $1.23666 \times 3001 = 3711.21666$. Hmm... no good. $1.23666 \times 301$ (hey, this might work) $ = 372.23466$. Nope. Okay, how about $1.23666 \times 30001 = 37101.03666$.

Lastly we try $1.23666 \times 300$ again, getting $370.998$ just like before. Okay, now try putting in as many trailing $6$s as your calculator will let you: $1.2366666666666666666666666666666$ $\times 300 =$ $370.99999999999999999999999999998$. One last verification: $371$ divided by $300$ gives $1.2366666666666666666666666666667$ on my calculator. So the answer is e.

The Short One
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How to solve the question:

multiply the number by two different powers of $10$

such that you can subtract them and obtain an integer.

$10^3a-10^2a=1236.\bar6 - 123.\bar6=1113$.

Therefore $900a=1113.$

Can you take it from here?

Note that $1113$ is divisible by $3$ (e.g., by the sum of digits test for divisibility by $3$) but not $9$ nor $100$.

J. W. Tanner
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You..... didnt repeat anything.

$a = 1.23666666666.....$

So $100a = 123.6666666666....$

And $100a - a = 123.666666666.....- 1.236666666..... = 123.66-1.23 = 122.43$.

So $a = \frac {122.43}{99}=\frac {12243}{9900}$

divide top and bottom by $33$ and we get $\frac {371}{300}$.

(you could multiplied by $10$ and gotten $10a - a = 12.3666666...- 1.236666 = 12.36 - 1.23 = 11.13$ to get $\frac {1113}{900}= \frac {371}{300}$ and avoid dividing by $11$.)

.....

Alternatively.

$a = 1.236666666....$ so $100a = 123.666666.....$

Now we know $0.66666..... = \frac 23$ (by the same method. $a = 0.6666...$ so $10a -a = 6.66666... - 0.666666....=6$ so $9a =6$ so $a =\frac 69 =\frac 23$.)

So $100a = 123\frac 23= \frac {271}{3}$ so $a = \frac {271}{300}$.

If we allow "abuse of notation" and allow ourselves to write:

$a= 1.236 = 1.23\frac 23$ (This is WRONG but ... "you know what I mean.....")

$a = 1.23\frac 23 = 1\frac {23\frac 23}{100}$ (This is NOT wrong)

$= 1\frac {23*3 + 2}{300} = 1\frac {71}{300} = \frac {371}{300}$.

fleablood
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$1.23\overline 6 = 1.23 + .00\overline 6= 1 + \frac{23}{100}+\frac{\frac{2}{3}}{100}= \frac{300}{300} + \frac{69}{300}+\frac{2}{300}= \frac{371}{300}$

CopyPasteIt
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